Limited feedback method and apparatus for two-way wireless relaying channels with physical network coding

ABSTRACT

A limited feedback method and apparatus for two-way relay channels with physical network coding are disclosed. An embodiment of the invention provides a method of providing parameters as feedback to two terminals according to channel conditions by a relay for two-way communication in a two-way relaying system using PNC (physical network coding). This method includes: (a) quantizing phase difference information of two-way channels of the two terminals in consideration of whether or not a ratio of a minimum distance between constellation points is periodic according to modulation level; and (b) transmitting feedback information, which contains at least one of the phase difference information and the power control information of the two terminals, as a limited number of bits to the two terminals. Here, the number of bits for the phase difference information or the number of bits for the power control information is determined according to the modulation level.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of Korean Patent Application No.10-2012-0118018, filed with the Korean Intellectual Property Office onOct. 23, 2012, the disclosure of which is incorporated herein byreference in its entirety.

BACKGROUND

1. Technical Field

The present invention relates to a limited feedback method and apparatusfor two-way relay channels with physical network coding, moreparticularly to a method and apparatus for providing phase informationand power allocation information as feedback for two terminals with awireless relay communicating bi-directionally.

2. Description of the Related Art

Wireless network coding has been a subject of intense research over thepast several years in the area of cooperative relaying systems.

Recently, a two-way relaying system using physical network coding (PNC)was introduced.

As illustrated in FIGS. 1A and 1B, the protocol may be divided into amultiple access (MA) stage, in which the relay node (relay) 100 receivesdata simultaneously from two end nodes (a first terminal 102 and asecond terminal 104), and a broadcast (BC) stage, in which the relay 100simultaneously broadcasts the data received from the first terminal 102to the second terminal 104 and the data received from the secondterminal 104 to the first terminal 102.

Various systems for PNC have been proposed, some of which are as shownbelow.

S. Katti, H. Rahul, W. Hu, D. Katabi, and M. Medard, “XORs in the air:practical wireless network coding,” in Proc. Conf. Applications,Technol., Architect., and Protocols Comput. Commun., September 2006,introduces the Exclusive-OR (XOR) operation, which scheme entails asimple operation for PNC. This operation, however, suffers fromperformance degradation in the system due to noise from other users.

S. Zhang, S. C. Liew, and P. P. Lam, “Physical-Layer Network Coding,” inProc. ACM MOBICOM, September 2006, considers the modulo operationinstead of the XOR technique, in order to improve system performance.Assuming perfect synchronization, the modulo operation for PNC canachieve almost the same bit error rate (BER) performance as one-waysystems with twice as much throughput.

Y.-T. Kim, M. Park, K.-J. Lee, and I. Lee, “Linear Precoding Designbased on the Minimum Distance for Two-Way MIMO Physical Network CodingSystems,” in Proc. IEEE Globecom, December 2011, shows that employingthe modulo operation and using precoding to make the channels of bothterminals equal makes it possible to obtain optimal performance in termsof the minimum distance for a two-way relaying system.

T. Koike-Akino, P. Popovski, and V. Tarokh, “Optimized constellationsfor two-way wireless relaying with physical network coding,” IEEEJournal on Selected Areas in Communications, vol. 27, pp. 773-787, June2009, proposes determining network coding functions and relay mappersfor all instant channel conditions to maximize the minimum distance. Thepaper introduces a scheme in which the network coding functions andrelay mappers selected by the relay are sent to the terminals by way ofa limited number of feedback bits. Although this scheme exhibits goodperformance compared to the XOR system, there is the drawback that everynode needs to know not only the numerous network coding functions andrelay mappers but also very sophisticated selection criteria subject tothe channel condition.

Considering the related art, it may be preferable to use a modulooperation for PNC in a two-way relaying system, but it is needed tolower the complexity compared to the existing limited feedback scheme.

SUMMARY

An objective of the invention is to provide a limited feedback methodand apparatus for physical network coding in two-way relaying channelsthat can provide high performance with low complexity.

To achieve the objective above, an embodiment of the invention providesa method of providing parameters as feedback to two terminals accordingto channel conditions by a relay for two-way communication in a two-wayrelaying system using PNC (physical network coding). This methodincludes: (a) quantizing phase difference information of two-waychannels of the two terminals in consideration of whether or not a ratioof a minimum distance between constellation points is periodic accordingto modulation level; and (b) transmitting feedback information, whichcontains at least one of the phase difference information and the powercontrol information of the two terminals, as a limited number of bits tothe two terminals. Here, the number of bits for the phase differenceinformation or the number of bits for the power control information isdetermined according to the modulation level.

If the modulation level is QPSK (quadrature phase shift keying), saidstep (a) can include reducing the number of bits for the phasedifference information by one bit and generating codebook candidateswithin a range of [0, π], in consideration of a period for the ratio ofthe minimum distance being π, and determining the phase differenceinformation to be a codebook candidate satisfying a particular criterionfrom among the codebook candidates.

If the modulation level is QPSK, it is possible for the feedbackinformation might not to include the power allocation information.

If the modulation level is 16-QAM (quadrature amplitude modulation), 1bit of the feedback information can contain information for identifyinga channel gain size of the two terminals.

The remaining bits of the feedback information can contain power controlinformation for controlling a power of a terminal having a greaterchannel gain.

The power of the terminal having a greater channel gain can bedetermined by the formula shown below:

$P_{M} = \frac{P_{M_{c}}}{\gamma^{2}\left( {P_{M_{c}},P_{m_{c}}} \right)}$

where P_(M) is the power of the terminal having a greater channel gain,P_(MC) is a power constraint for the terminal having a greater channelgain, P_(mC) is a power constraint for a terminal having a smallerchannel gain, and γ(P_(M), P_(m)) is power control information.

If the power control information is between 1 and 4, the remaining bitscan be selected from a codebook such that the power control informationapproaches 1.

The quantization model of the power control information can be expressedby the formula shown below:

γ(P _(M) _(C) , P _(m) _(C) )= γ _(i) if α_(i−1)≦γ(P _(M) _(C) , P _(m)_(C) )<α_(i)

where i=1, 2, . . . , 2^(FB) ^(P) ⁻¹, and FB₀ is the number of feedbackbits for θ_(C).

If γ(P_(M) _(C) , P_(m) _(C) )≧α₂ _(FB) _(P) ⁻¹ (α₂ _(FB) _(P) ⁻¹ ≦4) orα₀≦γ(P_(M) _(C) )<α₁, then the first quantization level can be set to{circumflex over (γ)}₁=1.

{circumflex over (γ)}_(i) and α_(i) can be determined as values whichsatisfy the three conditions shown below:

$\begin{matrix}{{{{\overset{\_}{\gamma}}_{i} - {a_{i - 1}\text{:}\mspace{14mu} a_{i}} - {\overset{\_}{\gamma}}_{i}} = {{2\text{:}\mspace{14mu} 3\mspace{14mu} {for}\mspace{14mu} i} = 2}},3,\ldots \mspace{14mu},{2^{N}.}} & 1 \\{\frac{5\left( {a_{1} - a_{0}} \right)}{3\; {\overset{\_}{\gamma}}_{1}} = {\frac{a_{2} - a_{1}}{{\overset{\_}{\gamma}}_{2}} = {\frac{a_{3} - a_{2}}{{\overset{\_}{\gamma}}_{3}} = {\ldots = {\frac{a_{N} - a_{N - 1}}{{\overset{\_}{\gamma}}_{N}}.}}}}} & 2 \\{{a_{1} - {a_{0}\text{:}\mspace{14mu} 4} - a_{N}} = {1\text{:}\mspace{14mu} 2.}} & 3\end{matrix}$

where N

2^(FB) ^(P) ⁻¹.

Another aspect of the invention provides a two-way communication relayconfigured to provide an optimal precoder as feedback to two terminalsin a two-way relaying system using PNC (physical network coding). Thistwo-way communication relay includes: a phase difference determiner unitconfigured to quantize phase difference information of two-way channelsof the two terminals in consideration of a period for a ratio of aminimum distance between constellation points; a transmit powerdeterminer unit configured to quantize power allocation information ofthe two terminals according to modulation level; and a transmitter unitconfigured to transmit feedback information, which contains at least oneof the phase difference information and the power control information ofthe two terminals, as a limited number of bits to the two terminals,where the number of bits for the phase difference information or thenumber of bits for the power control information is determined accordingto the modulation level.

An embodiment of the invention makes it possible to provide feedback bywhich to obtain an optimal minimum distance using a smaller number ofbits according to modulation level, thereby providing the advantage ofhigher performance with lower complexity.

Additional aspects and advantages of the present invention will be setforth in part in the description which follows, and in part will beobvious from the description, or may be learned by practice of theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A and FIG. 1B illustrate a two-way relaying system for PNC.

FIG. 2 is a block diagram of a relay according to an embodiment of theinvention.

FIG. 3 represents the throughput of PNC systems using QPSK inNakagami-Rice fading channels.

FIG. 4 represents the throughput of PNC systems using 16-QAM inNakagami-Rice fading channels.

DETAILED DESCRIPTION

As the present invention allows for various changes and numerousembodiments, particular embodiments will be illustrated in the drawingsand described in detail in the written description. However, this is notintended to limit the present invention to particular modes of practice,and it is to be appreciated that all changes, equivalents, andsubstitutes that do not depart from the spirit and technical scope ofthe present invention are encompassed in the present invention. Indescribing the drawings, like reference numerals are used for likeelements.

Certain embodiments of the invention will be described below in moredetail with reference to the accompanying drawings. To aid the overallunderstanding of the invention, like reference numerals are used forlike elements regardless of the figure number.

As illustrated in FIGS. 1A and 1B, an embodiment of the invention ispresented for a single-antenna relay system in which two terminals 102,104 each having one antenna communicates through a relay 100 having oneantenna.

The user terminals, end node A (first terminal 102) and end node B(second terminal 104), may transmit the symbols x_(A)=M(s_(A)) andx_(B)=M(s_(B)) using the M-QAM symbol mapper M(.), where s_(A) ε {0, 1,. . . , M} and s_(B) ε {0, 1, . . . , M}. Here, it is assumed thatE{|x_(i)|²}=1.

Then, the received signal of the relay 100 during the MA stage is asfollows:

y _(R)=√{square root over (P _(A))}h _(A) x _(A)+√{square root over (P_(B))}e ^(jθ) ^(c) h _(B) x _(B) +z _(R)   [Equation 1]

where P_(i) (i=A, B) denotes the transmit power of terminal i, θ_(C)indicates the phase difference between the two channels, h_(i)represents the channel between terminal i and the relay, and z_(R)represents noise at the relay.

In Equation 1, P_(i) and 0_(C) are the parameters of the feedbackinformation for the channels of the two terminals 102, 104.

The relay 100 may estimate s_(A) and s_(B) by maximum-likelihood (ML)detection as shown below.

$\begin{matrix}{\left( {{\hat{s}}_{A},{\hat{s}}_{B}} \right) = {\arg \; {\min\limits_{s_{A},s_{B}}\mspace{14mu} {{y_{R} - {\sqrt{P_{A}}h_{A}{M\left( s_{A} \right)}} - {\sqrt{P_{B}}^{j\; \theta_{c}}h_{B}{M\left( s_{B} \right)}}}}^{2}}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$

Based on the detected signals, physical network coding C(.) may beperformed to generate the network coding symbol s_(R)=C(ŝ_(A), ŝ_(B)).

At the BC stage, the relay 100 may transmit the network coding symbolobtained in the MA stage as x_(R)=M_(R)(s_(R)), using a relay mapperM_(R)(.). Then, the received signal at each terminal 102, 104 is asfollows:

y _(i)=√{square root over (P _(R))}h _(i) x _(R) +z _(i) for i=A and B

where P_(R) denotes the transmitted power of the relay.

Assuming channel reciprocity, it can be assumed that the channels of theMA and BC stages are the same.

Each terminal 102, 104 may extract the network coding symbol s_(R) andthen detect the signal transmitted by the other terminal by using itsown symbol s_(i) and the mapper C(.).

In the BC stage, since it is optimal for the relay 100 to use fullpower, P_(R)=P_(R) _(C) is used. Here, P_(R) _(C) represents the powerconstraint of the relay.

However, in the MA stage, the transmit powers P_(A) and P_(B) of therespective terminals 102, 104 and the phase difference θ_(C) may have tobe adjusted according to channel realizations.

This is because these may influence the minimum distance between theconstellation points of s_(A) and s_(B), which determines theperformance of the MA stage.

Here, the minimum distance can be expressed as follows:

$\begin{matrix}{\mspace{79mu} {{d_{\min}^{2}\overset{\bigtriangleup}{=}{\min\limits_{{C{({s_{A},s_{B}})}} \neq {C{({s_{A}^{\prime},s_{B}^{\prime}})}}}\mspace{14mu} {d^{2}\left( {s_{A},s_{A}^{\prime},s_{B},s_{B}^{\prime}} \right)}}}\mspace{79mu} {where}{{d^{2}\left( {s_{A},s_{A}^{\prime},s_{B},s_{B}^{\prime}} \right)} = {{{{\sqrt{P_{A}}h_{A}\Delta \; x_{A}} + {\sqrt{P_{B}}^{j\; \theta_{c}}h_{B}\Delta \; x_{B}}}}^{2}/\sigma^{2}}}\mspace{79mu} {and}\mspace{79mu} {{\Delta \; x_{i}} = {{{{M\left( s_{i} \right)} - {M\left( s_{i}^{\prime} \right)}}}.}}}} & \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack\end{matrix}$

Since the minimum distance of the MA stage cannot exceed either that oflink A or B, the following optimal value can be defined:

$\begin{matrix}{{d_{\min}^{2} \leq {\min\left( {\frac{P_{A}{h_{A}}^{2}\Delta_{\min}^{2}}{\sigma^{2}},\frac{P_{B}{h_{B}}^{2}\Delta_{\min}^{2}}{\sigma^{2}}} \right)}}\overset{\Delta}{=}d_{\min,{opt}}^{2}} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

where Δ_(min)

minΔx_(A)=minΔx_(B).

Y.-T. Kim et al. proved that choosing a modulo operation for the mapperC(.) and adjusting the power of each terminal and the phase can achievethe optimal performance which approaches the upper bound of the minimumdistance shown in Equation 4. Defining the inphase and the quadrature ofsymbol s_(i) as s_(iI) and s_(iQ), respectively, the network codingsymbol s_(R) may be obtained by the modulo operation as follows:

$\begin{matrix}\begin{matrix}{s_{Rk} = {C_{k}\left( {s_{Ak},s_{Bk}} \right)}} \\{= {{\left( {s_{Ak} + s_{Bk}} \right){mod}\sqrt{M}\mspace{14mu} {for}\mspace{14mu} k} = {I\mspace{14mu} {and}\mspace{14mu} Q}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

where C_(k)(.) represents the mapper for the inphase and the quadrature,(.)mod√{square root over (M)} indicates the modulo operation of size√{square root over (M)}. The modulo operation may generate the inphaseand the quadrature separately.

Here, the optimal values for transmit power P_(A) and P_(B) and phasedifference θ_(C) in terms of the minimum distance can be expressed asfollows:

$\begin{matrix}{{{P_{A} = P_{A_{C}}},{P_{B} = {{\frac{{h_{A}}^{2}}{{h_{B}}^{2}}P_{A_{C}}\mspace{14mu} {if}\mspace{14mu} P_{A_{C}}{h_{A}}^{2}} \leq {P_{B_{C}}{h_{B}}^{2}}}}}{{P_{A} = {\frac{{h_{B}}^{2}}{{h_{A}}^{2}}P_{B_{C}}}},{P_{B} = {{P_{B_{C}}\mspace{14mu} {if}\mspace{14mu} P_{A_{C}}{h_{A}}^{2}} > {P_{B_{C}}{h_{B}}^{2}}}}}{\theta_{C} = {\theta_{A} - \theta_{B}}}} & \left\lbrack {{Equation}\mspace{14mu} 6} \right\rbrack\end{matrix}$

where θ_(A) and θ_(B) are denoted as the channel phase of each terminal.With Equation 6, the optimal value for the minimum distance may beobtained by

${d_{\min,{opt}}^{2} = {\frac{\Delta_{\min}^{2}}{\sigma^{2}}{\min \left( {{P_{A}{h_{A}}^{2}},{P_{B}{h_{B}}^{2}}} \right)}}},$

and the two link channels become the same.

In order to obtain the optimal minimum distance, accurate values of thetransmit powers P_(A) and P_(B) and the phase difference 0_(C) may berequired as feedback. However, providing such information to theterminals in a limited number of bits, as is the case in an actualenvironment, may lead to degraded performance.

Thus, to resolve this issue, an embodiment of the invention provides anefficient feedback method for providing the transmit powers P_(A) andP_(B) and the phase difference θ_(C) for the cases in which themodulation level is QPSK and 16-QAM, based on Equation 6, which yieldsoptimal values assuming that the modulo operation in Equation 5 isadopted.

FIG. 2 is a block diagram illustrating the components of a relayaccording to an embodiment of the invention.

As illustrated in FIG. 2, a relay 100 according to an embodiment of theinvention can include a phase difference determiner unit 200, a transmitpower determiner unit 202, and a transmitter unit 204.

The phase difference determiner unit 200 may quantize the phasedifference information of the channels of the two terminals 102, 104communicating bi-directionally, in consideration of whether or not theratio of the minimum distance between constellation points is periodic,according to modulation level.

To be more specific, if the modulation level is QPSK, then the ratio ofthe minimum distance has a period of π, so that the phase differencedeterminer unit 200 may reduce one bit to design codebook candidateswithin the range of [0, π] and determine one of the designed codebookcandidates as the phase difference information.

If the modulation level is 16-QAM, the ratio of the minimum distance isnot periodic, so that the codebook candidates may be uniformly quantizedwithin [0, π], from which the phase difference information may bedetermined

The transmit power determiner unit 202 may determine the transmit powerallocation information for the two terminals performing two-waycommunication.

Similar to the case of phase difference, the transmit power determinerunit 202 can determine the power allocation information differentlyaccording to the modulation level.

For example, if the modulation level is QPSK, it may be preferable tohave both terminals transmitting with full power, so that there may beno separate power allocation.

However, in the case of 16-QAM, the transmit power determiner unit 202may determine the power allocation information in consideration of thechannel gains of the two terminals.

To be more specific, if the modulation level is 16-QAM, then thetransmit power determiner unit 202 may include information foridentifying the channel gain sizes of the two terminals in 1 bit fromamong the bits for the power allocation information. The remaining bitsmay include power control information for controlling the transmit powerof the terminal having the greater channel gain.

The phase difference information and power allocation informationdetermined as above may be transmitted to the two terminals in limitedbit numbers.

Thus, according to an embodiment of the invention, the information foroptimizing the minimum distance can be fed back with a small number ofbits according to modulation level.

That is, if the modulation level is QPSK, the bits for the phasedifference can be reduced by one bit, and if the modulation level is16-QAM, one bit can be allocated for the information identifying thechannel gain size, while the rest can be used for controlling thetransmit power. Thus, it is possible to improve performance with thegiven number of bits.

A more detailed description is provided below of the proceduresassociated with phase difference and power allocation.

(1) QPSK (Quadrature Phase Shift Keying)

First, for the feedback information of the phase difference θ_(C), auniform quantization within [0, 2π] may be needed.

However, it can be seen that the minimum distance ratio R

d_(min)/d_(min,opt) for θ_(C) has a period of π.

Therefore, according to an embodiment of the invention, one bit may bereduced, and the codebook may be designed within the range of [0, π] asfollows:

${CB}_{\theta} = \left\{ {0,\frac{\pi}{2^{{FB}_{\theta}}},\ldots \mspace{14mu},{\left( {2^{{FB}_{\theta}} - 1} \right)\frac{\pi}{2^{{FB}_{\theta}}}}} \right\}$

where FB_(θ) is the number of feedback bits for θ_(C). Then, thefeedback information for θ_(C) may be determined from among the codebookcandidates as the one satisfying the equation shown below:

${\hat{\theta}}_{C} = {\arg \; {\min\limits_{c_{i} \in {C\; B_{\theta}}}{{\left\{ {\left( {\theta_{A} - \theta_{B} + \frac{\pi}{2^{{FB}_{\theta} + 1}}} \right){mod}\; \pi} \right\} - \frac{\pi}{2^{{FB}_{\theta} + 1}} - c_{i}}}}}$

Next, there does not have to be any separate allocation of feedbackinformation related to transmit powers P_(A) and P_(B), because in QPSK,transmitting at full power (P_(A)=P_(A) _(C) , P_(B)=P_(B) _(C) ) alsomaximizes the minimum distance as was already proven.

(2) 16-QAM (Quadrature Amplitude Modulation)

For the feedback information of θ_(C) in 16-QAM, the ratio of minimumdistance is not periodic, in contrast to the case of QPSK. Therefore,the codebook for 16-QAM may be uniformly quantized in [0, 2π] as shownbelow:

${CB}_{\theta} = \left\{ {0,\frac{2\pi}{2^{{FB}_{\theta}}},\ldots \mspace{14mu},{\left( {2^{{FB}_{\theta}} - 1} \right)\frac{2\pi}{2^{{FB}_{\theta}}}}} \right\}$

The phase difference information fed back based on the above codebookmay be as follows:

${\hat{\theta}}_{C} = {\arg \; {\min\limits_{c_{i} \in {C\; B_{\theta}}}{{\theta_{A} - \theta_{B} - c_{i}}}}}$

Next, a description is provided of the quantization procedure fortransmit power allocation.

Assuming first that the phase difference θ_(C) was accuratelytransferred, the minimum distance can be expressed as follows:

$\begin{matrix}{d_{\min}^{2} = {\min\limits_{{C{({s_{M},s_{m}})}} \neq {C{({s_{M}^{\prime},s_{m}^{\prime}})}}}{\frac{P_{m}{h_{m}}^{2}}{\sigma^{2}}{{{\Delta \; x_{m}} + {{\gamma \left( {P_{M},P_{m}} \right)}\Delta \; x_{M}}}}^{2}}}} & \left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack\end{matrix}$

where the index of the terminal having the greater channel gain isrepresented as M

arg max_(iε{A,B})√{square root over (P_(iC))}|h_(i)|, the index of theterminal having the smaller channel gain is represented as m

arg min_(iε{A,B})√{square root over (P_(iC))}|h_(i)|, and the powercontrol information is defined as γ(P_(M), P_(m))

√{square root over (P_(M))}|h_(M)|/√{square root over (P_(m))}|h_(m)|.

In order to satisfy Equation 7, the terminal having the smaller channelgain may use maximum power (P_(m)=P_(m) _(C) ). On the other hand, forthe terminal having the greater channel gain, the transmit power may bereduced as follows:

$P_{M} = \frac{P_{M_{C}}}{\gamma^{2}\left( {P_{M_{C}},P_{m_{c}}} \right)}$

where γ(P_(M), P_(m)) represents the power control information. If thepower control information is 1, then the two channels become the same.Thus, in the limited feedback bit number FB_(P) for the transmit powerinformation, 1 bit may be used to indicate which of the terminals hasthe smaller channel gain. As described above, the terminal having thesmaller channel gain may perform transmission at maximum power. Theremaining bits may be selected from the codebook such that γ(P_(M),P_(m)) approaches 1.

A description is provided below of the design of an efficient codebookfor providing the feedback of γ(P_(M), P_(m)).

If γ(P_(M), P_(m)) is equal to 1 or is greater than or equal to 4, fullpower transmission may be employed by both of the two terminals 102, 104to obtain the maximum value for the minimum distance.

If γ(P_(M), P_(m)) is between 1 and 4, then the quantization model forγ(P_(M), P_(m)) can be expressed by the following equation:

γ(P _(M) _(C) , P _(m) _(C) )= γ _(i) if α_(i−1)≦γ(P _(M) _(C) , P _(m)_(C) )<α_(i)   [Equation 8]

for i=1, 2, . . . , 2^(FB) ^(P) ⁻¹. Here, 1 bit was used for determiningthe node M (the terminal having the greater channel gain) and m (theterminal having the smaller channel gain).

As described above, when γ(P_(M) _(C) , P_(m) _(C) )≧α₂ _(FB) _(P) ⁻¹(α₂ _(FB) _(P) ⁻¹ ≦4) or α₀≦γ(P_(M) _(C) )≦α₁, the first quantizationlevel may be set to γ ₁=1.

Next, γ _(i) and α_(i) may be determined to maximize the worst minimumdistance ratio.

According to an embodiment of the invention, γ _(i) and α_(i) may bedetermined such that the following three conditions are satisfied:

${{{1.\mspace{14mu} {\overset{\_}{\gamma}}_{i}} - {a_{i - 1}\text{:}a_{i}} - {\overset{\_}{\gamma}}_{i}} = {{2\text{:}3\mspace{14mu} {for}\mspace{14mu} i} = 2}},3,\ldots \mspace{14mu},2^{N}$${2.\mspace{14mu} \frac{5\left( {a_{1} - a_{0}} \right)}{3\; {\overset{\_}{\gamma}}_{1}}} = {\frac{a_{2} - a_{1}}{{\overset{\_}{\gamma}}_{2}} = {\frac{a_{3} - a_{2}}{{\overset{\_}{\gamma}}_{3}} = {\ldots = \frac{a_{N} - a_{N - 1}}{{\overset{\_}{\gamma}}_{N}}}}}$3.  a₁ − a₀:4 − a_(N) = 1:2

where N

2^(FB) ^(P) ⁻¹. The values of γ _(i) and α_(i) satisfying the threeconditions above can be obtained by computing the N number of equationsshown below:

k₁(3 k₁ + 2k₂ + 1) − k₂ = 0 k₁(3k₁ + 5k₂ + 2k₃ + 1) − k₃ = 0 ⋮${{k_{1}\left( {{3k_{1}} + {5{\sum\limits_{i = 2}^{N - 1}k_{i}}} + {2k_{N}} + 1} \right)} - k_{N}} = 0$${{9\; k_{1}} + {\sum\limits_{i = 2}^{N}k_{i}} - 3} = 0$${{where}\mspace{14mu} k_{1}}\overset{\Delta}{=}{{\frac{a_{1} - 1}{3}\mspace{14mu} {and}\mspace{14mu} k_{i}}\overset{\Delta}{=}{\frac{a_{i} - a_{i - 1}}{5}.}}$

Since there are N equations and N variables, k_(i) can be calculated bythe Newton Method, and γ _(i) and α_(i) can be ultimately computed usingk_(i). Thus, by using Equation 8, it is possible to feedback thetransmit power allocation of each terminal

FIG. 3 represents the throughput of PNC systems using QPSK inNakagami-Rice fading channels. Compared to the case of perfect feedbackinformation, it can be seen that the case of using 3 feedforward bitsyields almost the same performance results. Also, even when two bits areused according to an embodiment of the invention, a higher performanceis obtained compared to the existing adaptive-NC method [T. Koike-Akino]that uses 3 bits.

FIG. 4 represents the throughput of PNC systems using 16-QAM inNakagami-Rice fading channels. For comparison, the existing adaptive-NCmethod using 9 feedforward bits was selected. A method according to anembodiment of the invention used 5 bits for phase and 4 bits for powercontrol. It can be seen that the method according to an embodiment ofthe invention yields much higher performance even with a reduced numberof bits. Also, whereas the adaptive-NC method uses 300 network codes and15 QAM constellations at the relay, the method according to anembodiment of the invention adopts just a modulo operation and 16-QAM,and thus requires much lower complexity.

The embodiments of the invention described above are disclosed hereinfor illustrative purposes only. It is to be appreciated that variousmodifications, alterations, and additions can be made by those ofordinary skill in the art without departing from the technical spiritand scope of the invention, and that such modifications, alterations,and additions are encompassed by the scope of claims set forth below.

What is claimed is:
 1. A method of providing parameters as feedback totwo terminals according to channel conditions by a relay for two-waycommunication in a two-way relaying system using PNC (physical networkcoding), the method comprising: (a) quantizing phase differenceinformation of two-way channels of the two terminals in consideration ofwhether or not a ratio of a minimum distance between constellationpoints is periodic according to modulation level; and (b) transmittingfeedback information as a limited number of bits to the two terminals,the feedback information containing at least one of the phase differenceinformation and the power control information of the two terminals,wherein a number of bits for the phase difference information or anumber of bits for the power control information is determined accordingto the modulation level.
 2. The feedback method of claim 1, wherein, ifthe modulation level is QPSK (quadrature phase shift keying), said step(a) comprises: reducing the number of bits for the phase differenceinformation by one bit and generating codebook candidates within a rangeof [0, π] in consideration of a period for the ratio of the minimumdistance being π, and determining the phase difference information to bea codebook candidate satisfying a particular criterion from among thecodebook candidates.
 3. The feedback method of claim 2, wherein, if themodulation level is QPSK, the feedback information does not include thepower allocation information.
 4. The feedback method of claim 1,wherein, if the modulation level is 16-QAM (quadrature amplitudemodulation), 1 bit of the feedback information contains information foridentifying a channel gain size of the two terminals.
 5. The feedbackmethod of claim 4, wherein remaining bits of the feedback informationcontain power control information for controlling a power of a terminalhaving a greater channel gain.
 6. The feedback method of claim 5,wherein the power of the terminal having a greater channel gain isdetermined by a formula shown below:$P_{M} = \frac{P_{M_{C}}}{\gamma^{2}\left( {P_{M_{C}},P_{m_{c}}} \right)}$where P_(M) is the power of the terminal having a greater channel gain,P_(MC) is a power constraint for the terminal having a greater channelgain, P_(mC) is a power constraint for a terminal having a smallerchannel gain, and γ(P_(M), P_(m)) is power control information.
 7. Thefeedback method of claim 6, wherein the remaining bits are selected froma codebook such that the power control information approaches 1 if thepower control information is between 1 and
 4. 8. The feedback method ofclaim 6, wherein a quantization model of the power control informationis expressed by a formula shown below:γ(P _(M) _(C) , P _(m) _(C) )= γ _(i) if α_(i−1)≦γ(P _(M) _(C) , P _(m)_(C) )<α_(i) where i=1, 2, . . . , 2^(FB) ^(P) ⁻¹, and FB_(θ) is anumber of feedback bits for phase difference θ_(C).
 9. The feedbackmethod of claim 8, wherein a first quantization level is set to γ ₁=1 ifγ(P_(M) _(C) , P_(m) _(C) )≧α₂ _(FB) _(P) ⁻¹ (α₂ _(FB) _(P) ⁻¹ ≦4) orα₀≦γ(P_(M) _(C) )≦α₁.
 10. The feedback method of claim 9, wherein γ _(i)and α_(i) are determined as values which satisfy three conditions shownbelow:${{{1.\mspace{14mu} {\overset{\_}{\gamma}}_{i}} - {a_{i - 1}\text{:}a_{i}} - {\overset{\_}{\gamma}}_{i}} = {{2\text{:}3\mspace{14mu} {for}\mspace{14mu} i} = 2}},3,\ldots \mspace{14mu},2^{N}$${2.\mspace{14mu} \frac{5\left( {a_{1} - a_{0}} \right)}{3\; {\overset{\_}{\gamma}}_{1}}} = {\frac{a_{2} - a_{1}}{{\overset{\_}{\gamma}}_{2}} = {\frac{a_{3} - a_{2}}{{\overset{\_}{\gamma}}_{3}} = {\ldots = \frac{a_{N} - a_{N - 1}}{{\overset{\_}{\gamma}}_{N}}}}}$3.  a₁ − a₀:4 − a_(N) = 1:2${{where}\mspace{14mu} N}\overset{\Delta}{=}{2^{{FB}_{P} - 1}.}$ 11.A two-way communication relay configured to provide an optimal precoderas feedback to two terminals in a two-way relaying system using PNC(physical network coding), the two-way communication relay comprising: aphase difference determiner unit configured to quantize phase differenceinformation of two-way channels of the two terminals in consideration ofa period for a ratio of a minimum distance between constellation points;a transmit power determiner unit configured to quantize power allocationinformation of the two terminals according to modulation level; and atransmitter unit configured to transmit feedback information as alimited number of bits to the two terminals, the feedback informationcontaining at least one of the phase difference information and thepower control information of the two terminals, wherein a number of bitsfor the phase difference information or a number of bits for the powercontrol information is determined according to the modulation level. 12.The two-way communication relay of claim 11, wherein, if the modulationlevel is QPSK (quadrature phase shift keying), the phase differencedeterminer unit reduces the number of bits for the phase differenceinformation by one bit and generates codebook candidates within a rangeof [0, π] in consideration of a period for the ratio of the minimumdistance being π, and determines the phase difference information to bea codebook candidate satisfying a particular criterion from among thecodebook candidates.
 13. The two-way communication relay of claim 11,wherein, if the modulation level is 16-QAM (quadrature amplitudemodulation), 1 bit of the feedback information contains information foridentifying a channel gain size of the two terminals.
 14. The two-waycommunication relay of claim 13, wherein remaining bits of the feedbackinformation contain power control information for controlling a power ofa first terminal having a greater channel gain.
 15. The two-waycommunication relay of claim 14, wherein the power of the terminalhaving a greater channel gain is determined by a formula shown below:$P_{M} = \frac{P_{M_{C}}}{\gamma^{2}\left( {P_{M_{C}},P_{m_{c}}} \right)}$where P_(M) is the power of the terminal having a greater channel gain,P_(MC) is a power constraint for the terminal having a greater channelgain, P_(mC) is a power constraint for a terminal having a smallerchannel gain, and γ(P_(M), P_(m)) is power control information.